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Theorem undif4 3607
 Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undif4 ((AC) = → (A ∪ (B C)) = ((AB) C))

Proof of Theorem undif4
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 pm2.621 397 . . . . . . 7 ((x A → ¬ x C) → ((x A ¬ x C) → ¬ x C))
2 olc 373 . . . . . . 7 x C → (x A ¬ x C))
31, 2impbid1 194 . . . . . 6 ((x A → ¬ x C) → ((x A ¬ x C) ↔ ¬ x C))
43anbi2d 684 . . . . 5 ((x A → ¬ x C) → (((x A x B) (x A ¬ x C)) ↔ ((x A x B) ¬ x C)))
5 eldif 3221 . . . . . . 7 (x (B C) ↔ (x B ¬ x C))
65orbi2i 505 . . . . . 6 ((x A x (B C)) ↔ (x A (x B ¬ x C)))
7 ordi 834 . . . . . 6 ((x A (x B ¬ x C)) ↔ ((x A x B) (x A ¬ x C)))
86, 7bitri 240 . . . . 5 ((x A x (B C)) ↔ ((x A x B) (x A ¬ x C)))
9 elun 3220 . . . . . 6 (x (AB) ↔ (x A x B))
109anbi1i 676 . . . . 5 ((x (AB) ¬ x C) ↔ ((x A x B) ¬ x C))
114, 8, 103bitr4g 279 . . . 4 ((x A → ¬ x C) → ((x A x (B C)) ↔ (x (AB) ¬ x C)))
12 elun 3220 . . . 4 (x (A ∪ (B C)) ↔ (x A x (B C)))
13 eldif 3221 . . . 4 (x ((AB) C) ↔ (x (AB) ¬ x C))
1411, 12, 133bitr4g 279 . . 3 ((x A → ¬ x C) → (x (A ∪ (B C)) ↔ x ((AB) C)))
1514alimi 1559 . 2 (x(x A → ¬ x C) → x(x (A ∪ (B C)) ↔ x ((AB) C)))
16 disj1 3593 . 2 ((AC) = x(x A → ¬ x C))
17 dfcleq 2347 . 2 ((A ∪ (B C)) = ((AB) C) ↔ x(x (A ∪ (B C)) ↔ x ((AB) C)))
1815, 16, 173imtr4i 257 1 ((AC) = → (A ∪ (B C)) = ((AB) C))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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